Finite Group Databasebeta

← #37583997

Subgroups — #37583997

Order 24448.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259349C211yesmaximal, characteristic, complemented
2C22×D1528C211yesmaximal, characteristic, complemented
3#27133304C211yesmaximal, characteristic, complemented
4#27133297C211yesmaximal, characteristic, complemented
5#27133049C211yesmaximal, characteristic, complemented
6#27132917C211yesmaximal, characteristic, complemented
7#27132881C211yesmaximal, characteristic, complemented
8#27132857C211yesmaximal, characteristic, complemented
9#27132813C211yesmaximal, characteristic, complemented
10#27132811C211yesmaximal, characteristic, complemented
11#27132798C211yesmaximal, characteristic, complemented
12#27132796C211yesmaximal, characteristic, complemented
13#27132774C211yesmaximal, characteristic, complemented
14#27132734C211yesmaximal, characteristic, complemented
15#465×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic, complemented
17C22×D764C2211yescharacteristic
18C22×D764C2211yescharacteristic, complemented
19C2×D4×D191—21no—
20C2×D4×D191C2211yescharacteristic, complemented
21#20259349—24no—
22C2×#15690274C2211yescharacteristic, complemented
23C2×#15690274C2211yescharacteristic, complemented
24C2×D1528—24no—
25C2×D1528C2211yescharacteristic
26C2×D1528C2211yescharacteristic
27C2×D1528C2211yescharacteristic, complemented
28C2×D1528C2211yescharacteristic, complemented
29C2×#15690272C2211yescharacteristic, complemented
30C2×#15690272C2211yescharacteristic, complemented
31C2×#15690270C2211yescharacteristic, complemented
32#20259318C2211yescharacteristic, complemented
33#20259311C2211yescharacteristic, complemented
34#20259307—21no—
35#20259297—21no—
36#20259297C2211yescharacteristic, complemented
37#20259297C2211yescharacteristic, complemented
38#20259295—21no—
39#20259292—21no—
40#20259292—21no—
41#20259290—21no—
42#20259284C2211yescharacteristic, complemented
43#20259284C2211yescharacteristic, complemented
44C4×C22⋊D191C2211yescharacteristic
45C22⋊C4×D191—21no—
46Dic191⋊C8C2211yescharacteristic, complemented
47#20259242C2211yescharacteristic, complemented
48#20259242C2211yescharacteristic, complemented
49#20259242C2211yescharacteristic, complemented
50#20259242C2211yescharacteristic, complemented
51C22⋊(C191⋊C8)C2211yescharacteristic, complemented
52(C2×D191)⋊C8C2211yescharacteristic, complemented
53#20259232C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic, complemented
55C22×C8×C191C2211yescharacteristic, complemented
56C22⋊Q8×C191C2211yescharacteristic, complemented
57#121×C191C2211yescharacteristic
58#106×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63C23×D191—21no—
64#15690288—22no—
65#15690288—41no—
66C2×D764—22no—
67C2×D764—22no—
68C2×D764—21no—
69C2×D764—21no—
70C2×D764—21no—
71C2×D764—21no—
72C2×D764—21no—
73C2×D764—41no—
74C2×D764—41no—
75C2×D764C2311yescharacteristic
76C2×D764C2311yescharacteristic
77C2×D764C2311yescharacteristic
78C2×D764C2311yescharacteristic
79C2×C22⋊D191—21no—
80C2×C22⋊D191—21no—
81C2×C22⋊D191—21no—
82C2×C22⋊D191D411yescharacteristic
83D4×D191—22no—
84D4×D191—41no—
85D4×D191—44no—
86C2×C4×D191—21no—
87C2×C4×D191—21no—
88C2×C4×D191—21no—
89C2×C4×D191C2311yescharacteristic
90#15690274—22no—
91#15690274—41no—
92D1528—22no—
93D1528—22no—
94D1528—22no—
95D1528D414yescomplemented
96#15690272—22no—
97#15690272—41no—
98#15690270—22no—
99C2×C191⋊C8C2311yescharacteristic
100C2×C191⋊C8C2311yescharacteristic
101Dic191⋊C4D411yescharacteristic
102C22⋊Dic191D411yescharacteristic
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106(C2×D191)⋊C4—21no—
107(C2×D191)⋊C4—21no—
108(C2×D191)⋊C4—21no—
109(C2×D191)⋊C4D411yescharacteristic
110C4×Dic191C2311yescharacteristic
111C2×Q8×C191C2311yescharacteristic
112C2×D4×C191—21no—
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191C2311yescharacteristic
115SD16×C191—22no—
116C2×C8×C191—22no—
117C2×C8×C191C2311yescharacteristic
118C2×C8×C191C2311yescharacteristic
119C4⋊C4×C191—21no—
120C4⋊C4×C191C2311yescharacteristic
121C4⋊C4×C191C2311yescharacteristic
122C22⋊C4×C191—21no—
123C22⋊C4×C191—21no—
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—21no—
128C22×D191—42no—
129C22×D191—42no—
130C22×D191—42no—
131C22×D191—42no—
132C22×D191—42no—
133C22×D191—42no—
134C22×D191—81no—
135C22×D191—81no—
136C22×D191—81no—
137C22×D191C2×D411yescharacteristic
138D764—21no—
139D764—21no—
140D764—21no—
141D764—21no—
142D764—21no—
143D764—21no—
144D764—42no—
145D764—42no—
146D764—42no—
147D764—42no—
148D764—81no—
149D764C2×D412yes—
150D764C2×D412yescomplemented
151C22⋊D191—22no—
152C22⋊D191—42no—
153C22⋊D191—42no—
154C22⋊D191—81no—
155C2×Dic191C2×D411yescharacteristic
156C2×Dic191C2×D411yescharacteristic
157C2×Dic191C2×D411yescharacteristic
158C4×D191—21no—
159C4×D191—21no—
160C4×D191—42no—
161C4×D191—42no—
162C4×D191—81no—
163C191⋊C8—21no—
164C191⋊C8—21no—
165C23×C191—21no—
166C23×C191C2×D411yescharacteristic
167Q8×C191—21no—
168Q8×C191—21no—
169D4×C191—21no—
170D4×C191—21no—
171D4×C191—42no—
172C2×C4×C191—21no—
173C2×C4×C191—21no—
174C2×C4×C191—21no—
175C2×C4×C191—21no—
176C2×C4×C191—21no—
177C2×C4×C191C2×D411yescharacteristic
178C2×C4×C191C2411yescharacteristic
179C8×C191—21no—
180C8×C191C2×D412yescomplemented
181C22×C191—21no—
182C22×C191—21no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191AGL(1,ℤ/8)12yescomplemented
187C22×C191C22×D411yescharacteristic
188C2×D191—21no—
189C2×D191—22no—
190C2×D191—22no—
191C2×D191—21no—
192C2×D191—21no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—41no—
197C2×D191—41no—
198C2×D191—41no—
199C2×D191—41no—
200C2×D191—41no—
201C2×D191—41no—
202C2×D191—41no—
203C2×D191—84no—
204C2×D191—84no—
205C2×D191—84no—
206C4×C191—21no—
207C4×C191—41no—
208C4×C191—41no—
209C4×C191—41no—
210C4×C191C22×D411yescharacteristic
211C4×C191H(5,𝔽2)11yescharacteristic
212Dic191—21no—
213Dic191—21no—
214Dic191—21no—
215C2×C191—22no—
216C2×C191—81no—
217C2×C191D4211yescharacteristic
218C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C191#57611yescharacteristic
220D191—41no—
221D191—42no—
222D191—42no—
223D191—81no—
224D191—81no—
225C191#326411yescharacteristic, complemented, Sylow
226#3264—1911nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1911no—
228C22×D8—1911no—
229#546—1911no—
230D42—1911no—
231#496—1911no—
232#469—1911no—
233#465D19111yescharacteristic, complemented
234#463—1911no—
235#459—1911no—
236#450—1911no—
237#449—1911no—
238#447—1911no—
239#447—1911no—
240#442—1911no—
241#435—1911no—
242C2×#41—1911no—
243C22×D4—3821no—
244C22×D4—1911no—
245C22×D4—1911no—
246C22×D4—1911no—
247AGL(1,ℤ/8)—3824no—
248C2×SD16—1911no—
249C2×SD16—1911no—
250C2×SD16C2×D19111yescharacteristic, complemented
251C2×D8—3824no—
252C2×D8—1911no—
253C2×D8—1911no—
254C2×D8—1911no—
255C2×D8—1911no—
256C2×D8—1911no—
257C2×D8—1911no—
258C2×M16—1911no—
259C22×C8C2×D19111yescharacteristic, complemented
260#127—1911no—
261#124—1911no—
262#123—3821no—
263C22⋊Q8C2×D19111yescharacteristic, complemented
264#121—3821no—
265#121—3821no—
266#121—1911no—
267#121—1911no—
268#121C2×D19111yescharacteristic
269#120—3821no—
270#120—3821no—
271#120—3821no—
272C4×D4—1911no—
273C4×D4—1911no—
274C4×D4—1911no—
275C2×C22⋊C4—3821no—
276#106C2×D19111yescharacteristic, complemented
277C4⋊C8—1911no—
278#103C2×D19111yescharacteristic, complemented
279#102—1911no—
280#102—1911no—
281#102—1911no—
282#102—1911no—
283#102C2×D19111yescharacteristic, complemented
284C22⋊C8—1911no—
285C22⋊C8—1911no—
286#97—1911no—
287C24—3821no—
288C24—3821no—
289C24—3821no—
290#41—3822no—
291#41—7641no—
292C2×Q8C22×D19111yescharacteristic
293C2×D4—21no—
294C2×D4—3822no—
295C2×D4—3822no—
296C2×D4—3821no—
297C2×D4—3821no—
298C2×D4—3821no—
299C2×D4—3822no—
300C2×D4—3821no—
301C2×D4—3821no—
302C2×D4—3821no—
303C2×D4—3821no—
304C2×D4—3821no—
305C2×D4—1911no—
306C2×D4—1911no—
307C2×D4—1911no—
308C2×D4—1911no—
309C2×D4—1911no—
310C2×D4—7641no—
311C2×D4—7641no—
312C2×D4—7641no—
313C2×D4—7644no—
314C2×D4C22×D19111yescharacteristic
315C22×C4—3821no—
316C22×C4—3821no—
317C22×C4—3821no—
318C22×C4—1911no—
319C22×C4C22×D19111yescharacteristic
320SD16—22no—
321SD16—3822no—
322SD16—7641no—
323D8—3822no—
324D8—3822no—
325D8—3822no—
326D8—3822no—
327D8—1914no—
328D8—7641no—
329M16—3822no—
330C2×C8—22no—
331C2×C8—1911no—
332C2×C8—1911no—
333C2×C8C22×D19111yescharacteristic
334C2×C8C22×D19111yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—1911no—
337C4⋊C4C22×D19111yescharacteristic
338C4⋊C4C22×D19111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—3821no—
342C22⋊C4—3821no—
343C22⋊C4—3821no—
344C22⋊C4—3821no—
345C22⋊C4—3821no—
346C22⋊C4—3821no—
347C22⋊C4—1911no—
348C22⋊C4—1911no—
349C42—1911no—
350C23—21no—
351C23—3821no—
352C23—3821no—
353C23—3821no—
354C23—3821no—
355C23—1911no—
356C23—15281no—
357C23—15281no—
358C23—15281no—
359C23—7642no—
360C23—7642no—
361C23—7642no—
362C23—7642no—
363C23—7642no—
364C23—7642no—
365C23D4×D19111yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—42no—
371D4—3822no—
372D4—3821no—
373D4—3821no—
374D4—3821no—
375D4—3821no—
376D4—3821no—
377D4—3821no—
378D4—1912no—
379D4—1912no—
380D4—15281no—
381D4—15281no—
382D4—7642no—
383D4—7642no—
384D4—7642no—
385D4—7642no—
386D4—7642no—
387D4—7642no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—3821no—
394C2×C4—3821no—
395C2×C4—1911no—
396C2×C4—1911no—
397C2×C4—1911no—
398C2×C4—15281no—
399C2×C4—7642no—
400C2×C4—7642no—
401C2×C4D4×D19111yescharacteristic
402C2×C4C23×D19111yescharacteristic
403C8—21no—
404C8—3821no—
405C8—3821no—
406C8D4×D19112yescomplemented
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—3821no—
413C22—3821no—
414C22—3821no—
415C22—3822no—
416C22—3822no—
417C22—15284no—
418C22—15284no—
419C22—15284no—
420C22—7641no—
421C22—7641no—
422C22—7641no—
423C22—7641no—
424C22—7641no—
425C22—7641no—
426C22—7641no—
427C22—7641no—
428C22—7641no—
429C22—7641no—
430C22#2025934912yescomplemented
431C22C2×D4×D19111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—3821no—
437C4—3821no—
438C4—3821no—
439C4C2×D4×D19111yescharacteristic
440C4#2025936811yescharacteristic
441C2—22no—
442C2—81no—
443C2—15281no—
444C2—15281no—
445C2—7641no—
446C2—7642no—
447C2—7642no—
448C2#2713329711yescharacteristic
449C2C2×#2025934911yescharacteristic
450C2#2713346511yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#465×C191C2
socleC22×C191C22×D4
hypercenter#465D191
hypocenterC191#3264
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. C22×D1528C2semidirect
3. #27133304C2semidirect
4. #27133297C2semidirect
5. #27133049C2semidirect
6. #27132917C2semidirect
7. #27132881C2semidirect
8. #27132857C2semidirect
9. #27132813C2semidirect
10. #27132811C2semidirect
11. #27132798C2semidirect
12. #27132796C2semidirect
13. #27132774C2semidirect
14. #27132734C2semidirect
15. #465×C191C2semidirect
16. C2×#15690288C22semidirect
18. C22×D764C22semidirect
20. C2×D4×D191C22semidirect
22. C2×#15690274C22semidirect
23. C2×#15690274C22semidirect
27. C2×D1528C22semidirect
28. C2×D1528C22semidirect
29. C2×#15690272C22semidirect
30. C2×#15690272C22semidirect
31. C2×#15690270C22semidirect
32. #20259318C22semidirect
33. #20259311C22semidirect
36. #20259297C22semidirect
37. #20259297C22semidirect
42. #20259284C22semidirect
43. #20259284C22semidirect
46. Dic191⋊C8C22semidirect
47. #20259242C22semidirect
48. #20259242C22semidirect
49. #20259242C22semidirect
50. #20259242C22semidirect
51. C22⋊(C191⋊C8)C22semidirect
52. (C2×D191)⋊C8C22semidirect
53. #20259232C22semidirect
54. C2×SD16×C191C22semidirect
55. C22×C8×C191C22semidirect
56. C22⋊Q8×C191C22semidirect
58. #106×C191C22semidirect
59. #103×C191C22semidirect
60. #102×C191C22semidirect
95. D1528D4semidirect
150. D764C2×D4semidirect
180. C8×C191C2×D4semidirect
186. C22×C191AGL(1,ℤ/8)semidirect
225. C191#3264semidirect
233. #465D191semidirect
250. C2×SD16C2×D191semidirect
259. C22×C8C2×D191semidirect
263. C22⋊Q8C2×D191semidirect
276. #106C2×D191semidirect
278. #103C2×D191semidirect
283. #102C2×D191semidirect
406. C8D4×D191semidirect
430. C22#20259349semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#202593492
C22×D15282
#271333042
#271332972
#271330492
#271329172
#271328812
#271328572
#271328132
#271328112
#271327982
#271327962
#271327742
#271327342
#465×C1912
#3264191